134 Xili INTEGRATION 9. MEASURABLE FUNCTIONS (13.9.1) Let A be an integrable subset ofX. Then there exists a partition of A consisting of a sequence (Kn) of compact sets and a negligible set N. We define the sets Kn inductively as follows, using (13.7.9), (13.7.7), and (13.7.6): K! c A and /*(A - Ka) g 1; Kn <= A -"Q K, and if B > 1. Then the intersection N of the sets A — Q Kt is negligible, by (13.8.7). A subset of A is said to be measurable (with respect to /j) or ^-measurable if there exists a partition of A consisting of a sequence of compact sets and a negligible set. If K' and K are compact sets such that K' c K, then K ~ K' is integrable ((13.7.6) and (13.7.7)), and therefore, by virtue of (13.9.1), an equivalent statement is that A is the union of a sequence of compact sets and a negligible set. The same reasoning shows also that another equivalent statement is that A is the union of a sequence of integrable sets. A set A c X is said to be universally measurable if it is measurable with respect to every positive measure on X. (13.9.2) For a subset A of X to be measurable it is necessary and sufficient that A n K should be integrable, for each compact set K in X. The condition is necessary, by virtue of (13.8.7(ii)). It is sufficient, because X is the union of an increasing sequence (KM) of compact sets (3.18.3), and therefore A is the union of the sequence (A n Kn) of integrable sets. In particular, we see that the space X itself is measurable. (13.9.3) (i) The complement of a measurable set is measurable. (ii) Denumerable unions and denumerable intersections of measurable sets are measurable. (iii) All open sets and all closed sets are universally measurable. Assertion (i) follows from (13.9.2) and (13.7.6); assertion (ii) from (13.9.2) and (13.8.7); assertion (iii) from (13.9.2) and (13.7.7) for closed sets, and then by (i) for open sets.E <rM(A, , Aj)